Learn Finnish: Löysin tien kartan avulla. - I found the way with the help of the map.

The verb löytää (to find) describes a completed action with a definitive result. Therefore, it takes a total object. In standard active sentences, a singular total object takes the genitive-accusative form, which is tien here.

This lesson teaches the postposition nojalla (on the basis of). Could I say kartan nojalla instead of kartan avulla?

No, that would sound strange. While both are postpositions that take the genitive case, avulla (derived from apu, help) is used for practical tools, methods, or assistance. Nojalla is much more abstract and means 'by virtue of' or 'on the authority of', almost exclusively used with things like rules, facts, or laws (for example, lain nojalla).

Is avulla just the noun apu in a different case?

Exactly! It is the adessive case (-lla) of apu (help), literally meaning 'on/with help'. Many Finnish postpositions are just fossilized case forms of nouns. Because it still acts somewhat like a noun, it can also take possessive suffixes, such as in the phrase sinun avullasi (with your help).

Does avulla have to come after kartan, or can the word order be flipped?

It must come after. Avulla, just like the lesson's target words ohella and nojalla, is strictly a postposition. While a handful of Finnish adpositions can go before or after the noun, these abstract genitive-governing ones firmly belong after the noun they modify.